>>13940405Sure, that's a construction you can make. But here's the kicker (going off our example where x is the largest number, and a is any other element):
x plus a equals b, and b is not in our set
x plus a is EQUIVALENT to b, and b IS in our set
By constructing a ring, it seems like we are closing our set under addition, but a ring need only use equivalence, not equality. The classic example is modular arithmetic, where 2 does not EQUAL 5, but 2 is EQUIVALENT to 5 (modular 3).
Technical bullshit you may or may not already know, skip it if you don't care:
Equality (=) is a special kind of equivalence (~), an equivalence is any binary relation (relation between two things) that has three properties: reflexive (a~a), symmetric (a~b iff b~a), and transitive (a~b, b~c -> a~c). This is very general, and many kinds of relations can be defined so long as they have these properties. Equality, however, also has the property that it is antisymmetric (if a and b are different and a~b, then b~a must not be true), which is sometimes necessary for a proof to still hold. So if you rewrite the criteria for "closed under addition" to use equivalence rather than equality, it suddenly means something entirely different, not nearly as strict, and a lot of the properties we normally see to be true start to loosen up and fail. Hope this helps.
>t. someone who has not taken group theory yet and does not know what hes talking about, pls correct me if im wrong 
